1. Simplify combining like terms:
(i) 21b – 32 + 7b – 20b
Solution:
= (21b + 7b – 20b) – 32
= b (21 + 7 – 20) – 32
= b (28 – 20) – 32
= b (8) – 32
= 8b – 32
(ii) – z2 + 13z2 – 5z + 7z3 – 15z
Solution:
= 7z3 + (-z2 + 13z2) + (-5z – 15z)
= 7z3 + z2 (-1 + 13) + z (-5 – 15)
= 7z3 + z2 (12) + z (-20)
= 7z3 + 12z2 – 20z
(iii) p – (p – q) – q – (q – p)
Solution:
= p – p + q – q – q + p
= p – q
(iv) 3a – 2b – ab – (a – b + ab) + 3ab + b – a
Solution:
= 3a – 2b – ab – a + b – ab + 3ab + b – a
= 3a – a – a – 2b + b + b – ab – ab + 3ab
= a (1 – 1- 1) + b (-2 + 1 + 1) + ab (-1 -1 + 3)
= a (1 – 2) + b (-2 + 2) + ab (-2 + 3)
= a (1) + b (0) + ab (1)
= a + ab
(v) 5x2y – 5x2 + 3yx2 – 3y2 + x2 – y2 + 8xy2 – 3y2
Solution:
= 5x2y + 3yx2 – 5x2 + x2 – 3y2 – y2 – 3y2
= x2y (5 + 3) + x2 (- 5 + 1) + y2 (-3 – 1 -3) + 8xy2
= x2y (8) + x2 (-4) + y2 (-7) + 8xy2
= 8x2y – 4x2 – 7y2 + 8xy2
(vi) (3y2 + 5y – 4) – (8y – y2 – 4)
Solution:
= 3y2 + 5y – 4 – 8y + y2 + 4
= 3y2 + y2 + 5y – 8y – 4 + 4
= y2 (3 + 1) + y (5 – 8) + (-4 + 4)
= y2 (4) + y (-3) + (0)
= 4y2 – 3y
2. Add:
(i) 3mn, – 5mn, 8mn, – 4mn
Solution:
= 3mn + (-5mn) + 8mn + (- 4mn)
= 3mn – 5mn + 8mn – 4mn
= mn (3 – 5 + 8 – 4)
= mn (11 – 9)
= mn (2)
= 2mn
(ii) t – 8tz, 3tz – z, z – t
Solution:
= t – 8tz + (3tz – z) + (z – t)
= t – 8tz + 3tz – z + z – t
= t – t – 8tz + 3tz – z + z
= t (1 – 1) + tz (- 8 + 3) + z (-1 + 1)
= t (0) + tz (- 5) + z (0)
= – 5tz
(iii) – 7mn + 5, 12mn + 2, 9mn – 8, – 2mn – 3
Solution:
= – 7mn + 5 + 12mn + 2 + (9mn – 8) + (- 2mn – 3)
= – 7mn + 5 + 12mn + 2 + 9mn – 8 – 2mn – 3
= – 7mn + 12mn + 9mn – 2mn + 5 + 2 – 8 – 3
= mn (-7 + 12 + 9 – 2) + (5 + 2 – 8 – 3)
= mn (- 9 + 21) + (7 – 11)
= mn (12) – 4
= 12mn – 4
(iv) a + b – 3, b – a + 3, a – b + 3
Solution:
= a + b – 3 + (b – a + 3) + (a – b + 3)
= a + b – 3 + b – a + 3 + a – b + 3
= a – a + a + b + b – b – 3 + 3 + 3
= a (1 – 1 + 1) + b (1 + 1 – 1) + (-3 + 3 + 3)
= a (2 -1) + b (2 -1) + (-3 + 6)
= a (1) + b (1) + (3)
= a + b + 3
(v) 14x + 10y – 12xy – 13, 18 – 7x – 10y + 8xy, 4xy
Solution:
= 14x + 10y – 12xy – 13 + (18 – 7x – 10y + 8xy) + 4xy
= 14x + 10y – 12xy – 13 + 18 – 7x – 10y + 8xy + 4xy
= 14x – 7x + 10y– 10y – 12xy + 8xy + 4xy – 13 + 18
= x (14 – 7) + y (10 – 10) + xy(-12 + 8 + 4) + (-13 + 18)
= x (7) + y (0) + xy(0) + (5)
= 7x + 5
(vi) 5m – 7n, 3n – 4m + 2, 2m – 3mn – 5
Solution:
= 5m – 7n + (3n – 4m + 2) + (2m – 3mn – 5)
= 5m – 7n + 3n – 4m + 2 + 2m – 3mn – 5
= 5m – 4m + 2m – 7n + 3n – 3mn + 2 – 5
= m (5 – 4 + 2) + n (-7 + 3) – 3mn + (2 – 5)
= m (3) + n (-4) – 3mn + (-3)
= 3m – 4n – 3mn – 3
(vii) 4x2y, – 3xy2, –5xy2, 5x2y
Solution:
= 4x2y + (-3xy2) + (-5xy2) + 5x2y
= 4x2y + 5x2y – 3xy2 – 5xy2
= x2y (4 + 5) + xy2 (-3 – 5)
= x2y (9) + xy2 (- 8)
= 9x2y – 8xy2
(viii) 3p2q2 – 4pq + 5, – 10 p2q2, 15 + 9pq + 7p2q2
Solution:
= 3p2q2 – 4pq + 5 + (- 10p2q2) + 15 + 9pq + 7p2q2
= 3p2q2 – 10p2q2 + 7p2q2 – 4pq + 9pq + 5 + 15
= p2q2 (3 -10 + 7) + pq (-4 + 9) + (5 + 15)
= p2q2 (0) + pq (5) + 20
= 5pq + 20
(ix) ab – 4a, 4b – ab, 4a – 4b
Solution:
= ab – 4a + (4b – ab) + (4a – 4b)
= ab – 4a + 4b – ab + 4a – 4b
= ab – ab – 4a + 4a + 4b – 4b
= ab (1 -1) + a (4 – 4) + b (4 – 4)
= ab (0) + a (0) + b (0)
= 0
(x) x2 – y2 – 1, y2 – 1 – x2, 1 – x2 – y2
Solution:
= x2 – y2 – 1 + (y2 – 1 – x2) + (1 – x2 – y2)
= x2 – y2 – 1 + y2 – 1 – x2 + 1 – x2 – y2
= x2 – x2 – x2 – y2 + y2 – y2 – 1 – 1 + 1
= x2 (1 – 1- 1) + y2 (-1 + 1 – 1) + (-1 -1 + 1)
= x2 (1 – 2) + y2 (-2 +1) + (-2 + 1)
= x2 (-1) + y2 (-1) + (-1)
= -x2 – y2 -1
3. Subtract:
(i) –5y2 from y2
Solution:
= y2 – (-5y2)
= y2 + 5y2
= 6y2
(ii) 6xy from –12xy
Solution:
= -12xy – 6xy
= – 18xy
(iii) (a – b) from (a + b)
Solution:
= (a + b) – (a – b)
= a + b – a + b
= a – a + b + b
= a (1 – 1) + b (1 + 1)
= a (0) + b (2)
= 2b
(iv) a (b – 5) from b (5 – a)
Solution:
= b (5 -a) – a (b – 5)
= 5b – ab – ab + 5a
= 5b + ab (-1 -1) + 5a
= 5a + 5b – 2ab
(v) –m2 + 5mn from 4m2 – 3mn + 8
Solution:
= 4m2 – 3mn + 8 – (- m2 + 5mn)
= 4m2 – 3mn + 8 + m2 – 5mn
= 4m2 + m2 – 3mn – 5mn + 8
= 5m2 – 8mn + 8
(vi) – x2 + 10x – 5 from 5x – 10
Solution:
= 5x – 10 – (-x2 + 10x – 5)
= 5x – 10 + x2 – 10x + 5
= x2 + 5x – 10x – 10 + 5
= x2 – 5x – 5
(vii) 5a2 – 7ab + 5b2 from 3ab – 2a2 – 2b2
Solution:
= 3ab – 2a2 – 2b2 – (5a2 – 7ab + 5b2)
= 3ab – 2a2 – 2b2 – 5a2 + 7ab – 5b2
= 3ab + 7ab – 2a2 – 5a2 – 2b2 – 5b2
= 10ab – 7a2 – 7b2
(viii) 4pq – 5q2 – 3p2 from 5p2 + 3q2 – pq
Solution:
= 5p2 + 3q2 – pq – (4pq – 5q2 – 3p2)
= 5p2 + 3q2 – pq – 4pq + 5q2 + 3p2
= 5p2 + 3p2 + 3q2 + 5q2 – pq – 4pq
= 8p2 + 8q2 – 5pq
4. (a) What should be added to x2 + xy + y2 to obtain 2x2 + 3xy?
Solution:
Let us assume b be the required term
Then,
b + (x2 + xy + y2) = 2x2 + 3xy
b = (2x2 + 3xy) – (x2 + xy + y2)
b = 2x2 + 3xy – x2 – xy – y2
b = 2x2 – x2 + 3xy – xy – y2
b = x2 + 2xy – y2
(b) What should be subtracted from 2a + 8b + 10 to get – 3a + 7b + 16?
Solution:
Let us assume p be the required term
Then,
2a + 8b + 10 – p = -3a + 7b + 16
p = (2a + 8b + 10) – (-3a + 7b + 16)
p = 2a + 8b + 10 + 3a – 7b – 16
p = 2a + 3a + 8b – 7b + 10 – 16
p = 5a + b – 6
5. What should be taken away from 3x2 – 4y2 + 5xy + 20 to obtain – x2 – y2 + 6xy + 20?
Solution:
Let us assume b be the required term
Then,
3x2 – 4y2 + 5xy + 20 – b = -x2 – y2 + 6xy + 20
b = 3x2 – 4y2 + 5xy + 20 – (-x2 – y2 + 6xy + 20)
b = 3x2 – 4y2 + 5xy + 20 + x2 + y2 – 6xy – 20
b = 3x2 + x2 – 4y2 + y2 + 5xy – 6xy + 20 – 20
b = 4x2 – 3y2 – xy
6. (a) From the sum of 3x – y + 11 and – y – 11, subtract 3x – y – 11.
Solution:
First, we find the sum of 3x – y + 11 and – y – 11
= 3x – y + 11 + (-y – 11)
= 3x – y + 11 – y – 11
= 3x – y – y + 11 – 11
= 3x – 2y
Now, subtract 3x – y – 11 from 3x – 2y
= 3x – 2y – (3x – y – 11)
= 3x – 2y – 3x + y + 11
= 3x – 3x – 2y + y + 11
= -y + 11
(b) From the sum of 4 + 3x and 5 – 4x + 2x2, subtract the sum of 3x2 – 5x and
–x2 + 2x + 5.
Solution:
First, we find the sum of 4 + 3x and 5 – 4x + 2x2
= 4 + 3x + (5 – 4x + 2x2)
= 4 + 3x + 5 – 4x + 2x2
= 4 + 5 + 3x – 4x + 2x2
= 9 – x + 2x2
= 2x2 – x + 9 … [equation 1]
Then, we find the sum of 3x2 – 5x and – x2 + 2x + 5
= 3x2 – 5x + (-x2 + 2x + 5)
= 3x2 – 5x – x2 + 2x + 5
= 3x2 – x2 – 5x + 2x + 5
= 2x2 – 3x + 5 … [equation 2]
Now, we subtract equation (2) from equation (1)
= 2x2 – x + 9 – (2x2 – 3x + 5)
= 2x2 – x + 9 – 2x2 + 3x – 5
= 2x2 – 2x2 – x + 3x + 9 – 5
= 2x + 4
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